A note on strictly positive logics and word rewriting systems

TL;DR

This paper introduces a translation from word rewriting systems to strictly positive polymodal logics, demonstrating their generalization and providing examples of undecidable cases, along with a proof system and open questions in modal logic theory.

Contribution

It establishes a natural translation linking word rewriting systems to strictly positive polymodal logics, expanding understanding and providing new undecidability examples.

Findings

Strictly positive logics can be derived from word rewriting systems.

Undecidable strictly positive normal modal logics exist.

A deep inference proof system for these logics is formulated.

Abstract

We establish a natural translation from word rewriting systems to strictly positive polymodal logics. Thereby, the latter can be considered as a generalization of the former. As a corollary we obtain examples of undecidable strictly positive normal modal logics. The translation has its counterpart on the level of proofs: we formulate a natural deep inference proof system for strictly positive logics generalizing derivations in word rewriting systems. We also formulate some open questions related to the theory of modal companions of superintuitionistic logics that was initiated by L.L. Maximova and V.V. Rybakov.